Hey there!
Today I'm going to do something a little bit old-school and I'm going to be reviewing an
exam from 1866.
So, this is an arithmetic exam and it's from 152 years ago if my arithmetic is any good.
Now, this is called the Regents Questions and I believe it's from an exam from the state
of New York.
The Regents exams that are designed for I think graduating high school students to be
able to graduate and get their diploma.
So, to be able to prove you know, everything that you've sort of learned during your course
is there.
I believe the Regents exams still sort of exists today in some form but it's not the
same as it was back then.
This is an arithmetic exam and I think it's since been replaced by things like geometry,
algebra, calculus and many other different disciplines.
So regardless of what form the Regents exams look like these days, we're going to have
a look through these very old questions and just see what life was like back then for
a budding mathematician.
I'm going to go out on a hunch and assume that there are no calculators involved here,
so you're going to have to be doing all this by hand and it kind of makes sense that they
want to train you up to be good at arithmetic, to be good at things like bookkeeping or just
general day-to-day sums.
Now, I got these questions from a book which was published in 1880.
It's publicly available online and I'll link to it and it's just a database of all these
past problems going all the way back to the start.
So, follow along with that if you would like to and in my next video which will be uploaded
shortly after this one, I will actually work through in detail one of the more tricky questions
from this exam because in this video, we're not going to really work them out, just have
a look at what they're like.
So, let's get started.
So, here is page 1.
We have The Regents Questions: Arithmetic Examination 1.
November 8th, 1866.
Let's go through the problems.
Number 1: Write in figures each of the following numbers, add them and express in words their
sum.
So, we've got some numbers written down in words.
It would seem kind of easy to just write them down in digits but this is an 1866 exam after
all.
So, it's a little bit trickier than that.
They've given us fifty-six thousand and fourteen thousandths.
So, I guess they're talking about the places after the decimal point here, so you would
have like your tenths, hundredths, thousandths and tens of thousands digits.
So, the numbers are a little bit trickier.
We've got nineteen and nineteen hundredths, fifty-seven and forty-eight ten-thousandths,
twenty three thousand five, and four tenths and fourteen millionths.
So, a little bit trickier but I think it's still all right.
Number two; What is the difference between 3 8/4 plus 7 5/8 and 4 plus 2 8/4?
So this is a fractions problem.
We've got some mixed fractions here.
I think the rule is just going to be you need to add some fractions essentially and to do
that you'll need to, I guess, convert them to have a common denominator, so getting out
of mixed fractions into improper ones.
Question three: In multiplying by more than one figure, where is the first figure in each
partial product written, and why is it so written?
Now, this one seems a little bit confusing.
I guess I don't really know what they're saying.
As far as I can tell, they're talking about multiplying by more than one figure and to
me that means when you have say two numbers being multiplied together and each number
is like more than one digit in size and you do that thing where you do it column by column.
So you'd go 2 by 8, 16 carry 1, 1 by 8 plus that.
9 and then you do like the next one, the 10s, so that would be 4 and that would be 2 and
then you add these up.
That would be 3 3 6.
So I think that's what they're talking about but I guess I don't really know how to answer
the question; where is each partial product written.
Are these are the partial products?
Are they talking about carrying the one, uh, something like that.
Number four: If the divisor is 19, the quotient 37, and the remainder 11, what is the dividend?
So, the dividend is the number to be divided by something.
So, the dividend is divided by 19, the answer is 37 with a remainder of 11.
So, to work backwards and find what the dividend is, we do 19 times 37 plus 11.
Number five: What is the quotient of 65 bu. 1 pk and 3 qt. divided by 12?
So, these are a bunch of I guess units that I'm not familiar with.
I did a quick Google of them and I think it's called Peck and Bushel but yeah, I guess I
don't really know how many of each of these is in this one and likewise but you'd have
to convert them down to I guess the Quart divided by 12 and then maybe convert it back
into these types of units again.
Number six: Which of the fundamental operations (or ground rules) of arithmetic is employed
in reduction descending?
So, initially I had no idea what reduction descending is but I gave this one a quick
Google as well and I think it refers to when you're like going down in denomination.
Like so, if you have dollars and you want to convert it into cents.
If I had a dollar and I want to tell how many cents was in it, I would times it by a hundred.
So, I believe this one would be multiplication.
Number seven: In exchanging gold dust for cotton, by what weight would each be weighed?
This one seems a bit out of my range.
I guess I don't know exactly what they're saying but I feel like maybe they're on some
old-school scale and they're like needing to weigh gold dust and cotton against some
standard weight.
I'm not really sure.
Here's one that I think we'll be able to get.
What is the only even prime number?
The answer to that would be 2.
Number nine: How many weeks in (this many minutes)?
It about eight million.
I think this would be a process of dividing and probably a series of long division problems.
So you'd want to convert from minutes divide it by 60 to find out how many hours there
are, divided by 24 to get how many days, divided by seven days into weeks.
So, a series of division.
Hope you'll sharpen up your long division skills because mine are very rusty.
Number ten: To what term and division does the value of a common fraction correspond?
I don't really know what the terms in division are except that it could be like the dividend
or the divisor, the quotient or the remainder or something.
I'm gonna have to pass on that one though.
Number eleven: What is the product of a fraction multiplied by its denominator?
Well, if we had the fraction one-half and multiplied it by two, you just end up with
one or you just end up with whatever you started with on the top, so that would be ending up
with the numerator.
Number twelve: What is the rule for the multiplication of decimals?
Now, this one hasn't been outdated.
This rule still applies.
The rule is that the decimal point is placed in the product so that the number of decimal
places in the product is the sum of decimal places and what you're multiplying together.
So, for example if you're multiplying a number with two decimal places by a number with one
decimal place, your answer will have to have three decimal places in it.
That's still something that I remember doing in school.
Number thirteen: How is a common fraction reduced to a decimal?
I guess to convert a fraction to a decimal, what you're doing is you're dividing the numerator
by the denominator.
So, I guess that's all you can do, maybe do it by long division.
Number fourteen: What is ratio and how may it be expressed?
This one's a little difficult to put into words but I guess I would say that a ratio
is the number of times one value is like contained within the other.
15: If this amount of coal costs this amount of dollars, what will this amount of coal
costs?
I think this is another case of like converting this whole value into like the lowest denominator.
So like pounds is that?
I don't know, and then working out how much each pound costs.
Think converting this one into pounds too and working out its total cost.
Number 16: Find the cost of the several articles and the amount of the following bill.
This one is a little bit awkward.
I think it might got something to do with the rent or something.
So we've got A.P Jewett paying to Dr. Samuel Palmer for some amount feet of board at some
amount per M.
So, I guess this is literally square feet of space or that you're boarding and this
is either the cost per meter, like square meter per month.
I'm not sure.
I guess in essence, you'd have to understand the language here and what's common in terms
of these types of things but it probably comes down to just adding up these different articles.
17: What is the length of the side of a cubical box which contains 389,017 solid inches?
Excuse me, but this one seems pretty difficult to do without a calculator, like so much more
than the other ones.
I guess doing a cube root without a calculator is not something that I am well-versed in
but you would have to do it through either like long division and type sums or through
some sort of estimation and checking method.
Number eighteen: What is the present worth of the following note discounted at a bank,
and when will it become due?
So, we've got a nice little letter here; ninety days from date, for value received.
I promised to pay to the order of John Smith one hundred dollars at the Albany City National
Bank.
John Brown.
Forgive me if I'm mistaken, this one seems to just be asking how much it's worth and
it seems to be a hundred dollars paid 90 days from the date and this is the date here.
So, maybe this is some basic bookkeeping type skills although I would say these types of
word problems kind of do age this exam a little.
You really wouldn't see something written in the same way today.
Something that has not aged itself though, in fact has aged pretty well is things like
this where it's just using the numbers and the most basic sentences.
Maybe this one here involve 5/8 to the seventh power.
That's pretty much the same as what you'd see today except maybe the word involved would
be changed to find 5/8 or like so, 5/8 or something like that.
For this question though, it seems easy in essence.
You just need to take 5/8 times it by itself seven times, although when you do do it, it
turns out to get a little bit difficult on the algebra.
For example, I worked out what?
Five eighths to the sixth power would be...
So this would represent what you're going to have to do in your final sum to get the
seventh power, but to the sixth power, you're already dealing with fraction 15625 over 262144.
So, already you're getting quite high and you still need to take this and times it again
by 5 over 8.
So, I don't think that's necessarily like a walk in the park because even though you
know what to do, you do have to actually go into the details every time and it would be
a little bit of time to solve it.
Number 20: What is the square root of 0.0043046721?
Now, this question has me a little bit shook because just like the cube root question on
the previous page, this seems sort of absurd to be asking I think without the calculator,
but actually I guess it can be done and it's interesting to me to actually learn more about
how I would have done this.
So, in the next video which I'm going to upload shortly after this one, I'll take this problem
here and we'll see if we can solve it by hand.
Maybe we'll have a little bit of fun there, but I'm really interested to actually work
through how you would do that without the luxury of the calculator.
Maybe I'm too used to the luxury.
Number 21: We have an amount of sugar at a certain price.
It looks like it's like lost 12% in price, so how much was the whole cost.
For something like this, I guess in essence it's the same as what we still do these days.
If you want to find out how much it was worth before it depreciated, you would take the
total amount now divide it by 0.88 and that would tell you how much it was worth to start
with.
Number 22: A person owned five eighths of a mine and sold eight quarters of his interest
for this amount, what was the value of the entire mine?
I feel like this would have made sense to me if he was selling some sensible portion
of his share of the mine but it seems like he's selling... well, two of his share.
So, I don't know how you can actually sell more than what you have, that sort of seems
like to me.
I guess this must be like a multiplication and fractions somewhere but the wording is
so awkward I'd probably be a little bit tricked here.
Nearly at the end.
Number 23.
It's talking about, at a certain time, at a certain longitude, use that to work out
what time it will be at a different longitude.
This one actually seems interesting to me because I guess we take for granted like time
zones and longitude and I guess the relationship between.
But if you really do think about it, every hour of time zone difference would actually
correspond to a change of 15 degrees in the longitude.
So, I guess you can use that effect to work out here how the time is changing.
And finally another question using several different units here which are not [?] questions.
I mean, that's why it's hard to start with.
I guess converting it down into some unit that you choose and then dividing this price
by their number.
I don't know what potash is but okay.
I guess [?] this exam spooks me a little because I've gone through a lot of mathematics but
this - this paper is not really about mathematics.
Arithmetic is what it says.
It's really testing your long division, your multiplication, your basic skills like that
which can actually fall out of practice the further you go into maths.
It's sort of funny to talk with my friends about how bad we are sometimes at arithmetic
considering our qualifications in some of these really high level ideas.
You know, if we like go out to dinner together, it will take several of us to just work out
how to split the bill.
It's not like these are the skills that you actually really practice every day the further
you go, it gets more and more abstract from here.
But it makes sense.
This is 1866 and I guess even when I was in high school, our teachers in hindsight's sort
of lied to us and said "you've got to learn these skills because you're not going to just
have a calculator in your pocket everywhere you go".
It turns out that's - it turned out to be sort of true with smartphones and everything
but back in 1866 I guess that sentiment actually did ring true, that you didn't have a calculator
in your pocket.
I guess it raises the question of nowadays is arithmetic important.
Like eventually some of these skills have been phased out even in the Regents exam since
this first one and it's kind of my opinion that arithmetic is less and less important
to learn and to know and to have so much practice in because whilst I think it's still valuable
to know the foundations of what you're doing and to call yourself a mathematician without
understanding how to do long division, it's a little bit strange I admit but also there's
so much to learn and so much to do that really we only have time to focus on doing the important
things that are new.
But in having said that, probably around the world there are still some exams like this.
Some countries I think would focus more on making sure that someone who wants to call
themselves a mathematician can do these like square roots by hand.
I never had to prove that but I'm sure somewhere in the world, this is still the kind of thing
you're having to prove and I guess I'm not the judge of whether that's good or bad or
not, but respect to the people that could solve all of this exam no worries because
it's already what they've been practicing.
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